Measurement

Measurement is the assignment of numbers to objects in such a way that physical
relationships and operations among the objects correspond to arithmetic relationships and
operations among the numbers.

Both objects and relationships are mapped: All measurement is a form of modeling; it
embodies a primitive theory of how the objects work.

Different levels of measurement are defined by the number and kind of correspondences
that hold between the physical relations among objects and the arithmetic relations among
the scores.

Consider the example of measuring mass or weight. When we measure weight, two objects
that weigh the same are assigned the measured value, or score. Hence, the relationship
among objects is mirrored by equality of numbers.

Similarly, if two objects balance each other on a balance scale, our system of
measurement assigns the same weight to each. Hence the physical relationship of balance
maps onto (corresponds to) the mathematical relationship of equality.

Suppose two items x and y are placed together on left of a scale. The measured weight of
the two combined is equal to the sum of scores of each individual item

Suppose an object weighs 5 pounds. Then five of these objects will weight 5*5 = 25. So
multiplication of measured scores by a constant maps to the physical operation of putting
that many items on the scale.

But not all mathematical relationships among measured values have a counterpart in
physical operations. For example, the breakdown of an object's weight into its prime
multipliers doesn't say anything about the objects themselves. Similarly, if the weight of
one object happens to be the log of the weight of another, this does not imply any special
relationship among the objects.

The different levels of measurement are distinguished by which arithmetic operations on
the measured scores have counterparts in physical operations on the objects. To put it
another way, levels of measurement are distinguished by which arithmetic operations on the
scores are meaningful. In fact, in a deep sense, the theory of measurement is a theory of
meaningfulness -- this will become more clear as we go along! As an aside I would point
out that any time we talk about meaning, we are talking about some kind of mapping from
one system to another. To understand something is to build a model of it; it is to
translate it into different terms.

The rules that distinguish different scales of measurement also define the uniqueness
of measurement: the number and kind of alternative measurements that are valid. What i'm
talking about is the existence of things like alternative units. For example, if I can
measure height in meters (i'm 1.73 meters tall), then I can also measure it in centimeters
(173) or inches (68). These are all equally valid. This also will become clearer as we go
along.

Function Notation

A function is a mapping of the objects in one set to the objects in another. The two
sets can be the same. For example, the function Y = X2 is a mapping of real
numbers to real numbers. We use the notation f(x) = x2 to define a function.
You can think of f(x) as saying "a function of x". The right hand side of the
equation defines exactly what function of x it is.

If we are measuring the height of different persons, we might represent the measured
height of Steve as f(Steve), which might have the value f(Steve) = 68. To refer to the
height of someone who is only half as tall as Steve, we might write f(Joe) = 0.5*f(Steve).

Levels of Measurement

Nominal

In nominal measurement, we assign numeric scores in such a way that only equality of
scores has meaning for the attribute being measured. For example, consider measuring
weight on a nominal scale. Suppose that in my system of measurement, my weight is assigned
a "12". Now suppose that we compare my weight to yours. If you also were
assigned a 12, this would mean that we both had the same weight. But if you had any other
score, such as 15, there is nothing we could say: we could not say that you weigh
more than me. So in nominal scale measurement, the only physical property preserved or
captured by the numeric scores, is equality:

x is the same weight as y

if and only if

f(x) = f(y)

Note the tremendous lack of uniqueness of measurement scales: any other number system
that preserved that property would be just as good, and there are an infinite number of
them.

In order to tell whether two sets of measurements are the same, you need to recode both
of them so they use the same codes, then compare them: the fact that they initially have
different values doesn't mean anything.

Ordinal

In ordinal measurement, we assign numeric scores in such a way that not only equality
of scores but ordinality of scores have meaning for the attribute being measured. For
example, let us measure weight on an ordinal scale. Suppose that in my system of
measurement, my weight is assigned a "12". Now suppose we compare my weight to
yours. If you also were assigned a 12, this would mean that we both had the same weight.
So far, that's the same as in nominal measurement.

But if you had a weight of 24, this would mean not only that we have different weights,
but that you weigh more than I do, because 24 is bigger than 12. However, we can't say how
much more you weigh than I do. For example if A weights 12 and B weighs 16 and C weighs
24, we cannot say that the difference in weight between A and B is half of what the
difference between B and C is. All we know is that C weighs the most, A weighs the least,
and B is in between.

So in an ordinal scale, the only physical properties preserved or captured by the
measured scores is equality and ordinality:

x is the same weight as y

if and only if

f(x) = f(y)

x weighs more than y

if and only if

f(x) > f(y)

Again, any method of assigning numeric scores that satisfies these two rules is a valid
ordinal measurement. This means that ordinal measurements are unique only up to a
monotone transformation.

In order to tell whether two sets of measurements are the same, you need to rank order
both sets and then compare them: the fact that they initially have different values
doesn't mean anything.

Interval

In interval measurement, we assign numeric scores in such a way that not only equality
and ordinality of scores have meaning, but also the intervals between the scores. For
example, let us measure weight on an interval scale. Suppose that in my system of
measurement, my weight is assigned a "12". Now suppose we compare my weight to
yours. If you also were assigned a 12, this would mean that we both had the same weight.
But if you had a 24, this would mean that we not only had different weights, but that you
weigh more than I do. So far, this is the same as ordinal measurement. But here is the
different part: if A weighs 12 and B weighs 16 and C weighs 24, we can say that the
difference in weight between A and B is half of what the difference between B and C is.
The differences between scores have meaning now.

However, we still can't say that if f(A) is 12 and f(C) is 24, that C weighs twice as
much as A. As proof, consider temperature of two cities, measured in degrees fahrenheit.
city A is 80 degrees and city B is 40 degrees. We are tempted to say city A is twice as
hot. But suppose that instead we measure the temperature in centigrade. You know that to
get from fahrenheit to centigrade we subtract 32 and multiply by 5/9. So city A is 27
degress centigrade. City B is 4 degrees centigrade. It no longer appears to be twice as
hot: now it looks more like 7 times as hot. Now you know both centigrade and fahrenheit
are equally valid measuring scales of temperature, yet they are giving really different
impressions of the relative temperature of these two cities! The problem is that they are
interval-scale measures of temperature, and it is not meaningful to say that something is
twice something else when you measure it on an interval scale!

So in interval scale measurement, the only physical properties preserved or captured by
the measured scores are equality, ordinality, and interval ratios:

x is the same weight as y

if and only if

f(x) = f(y)

x weighs more than y

if and only if

f(x) > f(y)

the difference in weight bet.
A and C is k times as big as the
difference bet. A and B

if and only if

k = (f(a)-f(c))/(f(a)-f(b))

Any method of assigning numeric scores that satisfies these three rules is a valid
interval measurement, and this turns out to mean that interval measurements are unique up
to a linear transformation of the values. A linear transformation is one that looks like
this: g(x) = m*f(x) - b where m and b are constants. The transformation from fahrenheit to
centigrade is like this:

C = (F-32)*5/9,
C = 5/9F - 17.7.

In order to tell whether two sets of interval measurements are the same, you need to
standardize both of them and then compare them: the fact that they initially have
different values doesn't mean anything. So standardize an interval measurement, we
subtract mean value and divide by the standard deviation.

Ratio

In ratio-scale measurement, we assign numeric scores in such a way that not only
equality and ordinality and the intervals between the scores have meaning, but also ratios
of the scores. For example, let us measure weight on a ratio scale. Suppose that in my
system of measurement, my weight is assigned a "12". Now suppose we compare my
weight to yours. If you also were assigned a 12, this would mean that we both had the same
weight. But if you had a 24, this would mean that we not only had different weights, but
you weigh more than I do. Furthermore, if A weighs 12 and B weighs 16 and C weighs 24, we
can say that the difference in weight between A and B is half of what the difference
between B and C is. The difference between numbers has meaning. And, if your measured
weight is 24 and my measured weight is 12, then we can say that you weigh twice as much as
I do (finally!).

So in ratio scale measurement, the following properties are preserved by the measured
scores:

x is the same weight as y

if and only if

f(x) = f(y)

x weighs more than y

if and only if

f(x) > f(y)

the difference in weight bet.
A and C is k times as big as the
difference bet. A and B

if and only if

k = (f(a)-f(c))/(f(a)-f(b))

x weighs k times as much as y

if and only if

k = f(x)/f(y)

Any method of assigning numeric scores that satisfies these four rules is a valid
ratio-scale measurement, and this turns out to mean that ratio measurements are unique up
to a congruence or proportionality transformation of the values. A congruence
transformation is one that looks like this: g(x) = m*f(x) where m is a constant. For
example, we can get from inches to centimeters by this equation:

C = 2.54*I

In order to tell whether two sets of measurements are the same, you need to normalize
both of them and then compare them: the fact that they initially have different values
doesn't mean anything. To normalize ratio variables, we divide each value by the square
root of sum of squares of all values.